1. Vibrations and Waves
Chapter 13

2. Hooke’s Law When a spring is stretched, a restoring force is created.
The restoring force is proportional to the displacement
The restoring force is always directed toward the equilibrium point
The restoring force is always directed away from the displacement
F = -k ∙ x

3. Mass on a Vertical Spring
If the mass hangs motionlessly…
netF = ∑F = Fg + Fk = mg + kx = 0
(Fk : restoring force)

4. If both springs are identical, how would the displacement of A and B differ?

5. Simple Harmonic Motion
Motion that repeats in a regular pattern over and over again
The motion is driven by a net Force that obeys Hooke’s Law

6. Simple Harmonic Motion
At equilibrium:
The spring force and the mass’s acceleration become zero.
The speed reaches a maximum.
At maximum displacement:
The spring force and the mass’s acceleration reach a maximum.
The speed becomes zero.

7. Elastic Potential Energy
Energy stored in a stretched or compressed spring
PEs = ½ k x2
When you hang a mass on a spring, where does that spring PE come from?

8. Simple Harmonic Motion

9. Simple Harmonic Motion

10. Simple Harmonic Motion
Important Concepts
Amplitude (A): the maximum displacement of the object from its equilibrium position
Period (T): time it takes for the object to move through one complete cycle of motion
Frequency (f): the number of complete cycles completed in a unit of time (units s-1, Hz)

11. Motion of a Pendulum
Imagine a pendulum hanging straight down, in its neutral position
If the bob is displaced at a small angle (<15°), it will swing with SHM
What is the source of the restoring force?
Ft = -mg sinθ

12. Motion of a Pendulum
Describe the speed, force, acceleration as the bob crosses the equilibrium point, during SHM
max speed, zero restoring force, zero acceleration
Describe the speed, force, acceleration when the bob is at its maximum displacement from the equilibrium point.
zero speed, max restoring force, max acceleration
The period of a pendulum, at small angles, can be calculated

13. Motion of a Pendulum What is g? What factors effect the value of g?
If there is no friction, no non-conservative forces acting on the pendulum, what can be said about it?

14. Motion of a Pendulum

15. Waves
A wave is a disturbance that carries energy through matter or space.
It is a transfer of energy, not matter
How the energy is carried defines the type of wave.

16. Waves
mechanical waves propagate through a medium. (e.g. sound, water waves, earthquakes)
The medium is any material through which a wave travels (e.g. air, water, a spring, the Earth, or even people).

17. Waves
In a longitudinal wave, the particles in the medium move parallel to the direction of the wave (e.g. sound waves)

18. Waves
In a transverse wave, the particles in the medium move perpendicular to the direction of the wave (e.g. light waves)

19. Waves
In a surface wave, the particles typically move in circular paths in the medium.

20. Waves
You can find all three types of mechanical waves exhibited in earthquakes.

21. Waves
Basic wave anatomy
Crest
Trough
Amplitude
Wavelength

22. Waves
Wave speed (v) – How fast the wave is moving (the disturbance itself).
Speed depends on the medium.
Light: 3 x 108 m/s (300,000,000 m/s)
(671,000,000 miles/hour)
Sound: 343 meters/second (768 miles/hour)
Fastest: solids
liquids
Slowest: gasses

23. Wave characteristics
Frequency (f) – The number of waves passing by a point in a given period of time

24. Waves
Wave equations T: period v: wave speed, velocity λ: wavelength f: frequency

25. Waves

27. Wave interactions Interference
When more than one wave occupies the same space the results can vary depending upon how the waves match up

28. Wave interactions Constructive Interference
If the troughs and crests line up (even briefly), the resulting wave will have greater amplitude

29. Wave interactions Constructive Interference
If the troughs and crests line up (even briefly), the resulting wave will have greater amplitude

30. Wave interactions Destructive Interference
If the crest of one wave lines up with the trough of another wave, the resulting wave will have lower amplitude.

31. Wave interactions Destructive Interference
If the crest of one wave lines up with the trough of another wave, the resulting wave will have lower amplitude.

32. Waves
Waves on strings
The speed of a wave on a string is affected by tension force and the composition of the string
F: tension in the string
μ: linear density (mass per length)

33. Waves
Waves on Strings
When a wave encounters an object it can be reflected
Reflected waves in strings bounce off the surface of a rigid obstacle and are inverted, if the end of the string is fixed

34. Waves
Waves on Strings
When a wave encounters an object it can be reflected
if the end of the string is free to move at the boundary, reflected waves in strings bounce off the surface of a rigid obstacle and are not inverted